Originally Posted by rm -rf

Are your new cables filed cleanly on the ends, and fully seated in the shifter and frame bosses? It doesn't seem likely that it's a cable housing problem, but there could be housing installation problems where the cable drags a little.

From above posts:
derailleur hanger -- is it replaceable, or part of the steel frame?
very old chain?

Worn derailleur
My Campagnolo Veloce 10 speed rear derailleur wore out. The pivots were worn, so I could wiggle the shifter body in and out a little, with just light finger pressure. It had sloppy shifting.

The replacement was amazingly better, like getting a new drivetrain. It's pivots were redesigned, larger and set farther apart. The original design had small, pressed-in pins. This new derailleur lasted a long time.

I don't know if Shimano derailleurs wear out as quickly. I had maybe 15000 miles on the worn out one, I'd expect a lot more miles.